Convergence to type I distribution of the extremes of sequences defined by random difference equation
We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn=MnRn-1+q, n>=1, where R0 is arbitrary, (Mn) are iid copies of a non-degenerate random variable M, 0 0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (Rn) converge in distribution to a double-exponential random variable. This partially complements a result of deÂ Haan, Resnick, Rootzén, and deÂ Vries who considered extremes of the sequence (Rn) under the assumption that .
Volume (Year): 121 (2011)
Issue (Month): 10 (October)
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- Devroye, Luc & Fawzi, Omar, 2010. "Simulating the Dickman distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 242-247, February.
- de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
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